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\title{\huge{\heiti A Brief Summary of Quantum Computing }}
\author{\small{\kaishu Baining Shen}\\[2pt]
\small{Email:}
\url{3190101651@zju.edu.cn}
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\begin{flushleft}
\textbf{Abstract}: Quantum computing is a modern way of computing that is based on the science of quantum mechanics and its unbelievable phenomena. It is a beautiful combination of physics, mathematics, computer science and information theory. This paper starts with the difference of traditional computing and the quantum computing. Then it moves on to the fundamentals of quantum computing and the different categories of quantum computer. At last it exemplifies some applications of quantum computing. \\[8pt]
\textbf{Keywords}: quantum computing, quantum algorithm
\end{flushleft}
\section*{Traditional computing against quantum computing}
Today's computers are smaller, cheaper, faster, and even more powerful as compared to early computers that used to be huge, costly, and more power-consuming. Electronic circuits used in computers are getting smaller and smaller day by day. However, as these circuits become smaller, they will not obey the classical physical laws as they used to be. When we reduce the size of electronic circuits up to the size of an atom, there will be some quantum phenomena called “Quantum Tunneling”. Thus, there is a need for new computing other than current classical computing to put its state into some physical information rather than a circuit.


Quantum Computing is a new kind of computing based on Quantum mechanics that deals with the physical world that is probabilistic and unpredictable in nature. Quantum mechanics being a more general model of physics than classical mechanics give rise to a more general model of computing- quantum computing that has more potential to solve problems that cannot be solved by classical ones. To store and manipulate the information, they use their own quantum bits also called "Qubits" unlike other classical computers which are based on classical computing that uses binary bits 0 and 1 individually. The computers using such type of computing are known as "Quantum Computers". In such small computers, circuits with transistors, logic gates, and Integrated Circuits are not possible. Hence, it uses the subatomic particles like atoms, electrons, photons, and ions as their bits along with their information of spins and states. They can be superposed and can give more combinations. Therefore, they can run in parallel using memory efficiently and hence is more powerful. Quantum computing is the only model that could disobey the Church-Turing thesis and thus quantum computers can perform exponentially faster than classical computers.

\section*{Fundamentals in quantum computing}
\subsection*{Quantum bit}
Quantum Bit or Qubit is the fundamental unit of quantum information that represents subatomic particles such as atoms, electrons, etc. as a computer's memory while their control mechanisms work as a computer's processor. It can take the value of 0, 1, or both simultaneously. Production and management of qubits are tremendous challenges in the field of engineering. They acquire both, digital as well as analog nature which gives the quantum computer their computational power. Their analog nature indicates that quantum gates have no noise limit and their digital nature provides a norm to recover from this serious weakness. Therefore, the approach of logic gates and abstractions created for classical computing is of no use in quantum computing. Quantum computing may adopt ideas only from classical computing. But this computing needs its own method to overcome the variations of processing and any type of noise. It also needs its own strategy to debug errors and handle defects in design. 


We describe a qubit $ \left | \psi  \right \rangle $ in terms of two complex numbers $ \psi_0, \psi_1 \in \mathbb{C} $
$$ \ys{\psi} = \psi_0 \ys{0} + \psi_1 \ys{1} = \begin{pmatrix} \psi_0 \\  \psi_1 \end{pmatrix} $$
Both complex numbers $\psi_0$ and $\psi_1$ represent the amplitude of the corresponding state. Summation of the
squares of the absolute values of the amplitudes should be equal to 1. For a single qubit $ \ys{\psi} = \psi_0 \ys{0} + \psi_1 \ys{1} $ we have $ \left \langle \psi | \psi \right \rangle = \left\lvert \psi_0 \right\rvert ^2 + \left\lvert \psi_1 \right\rvert ^2 = 1 $


Informally, a complex superposition like $ \ys{\psi} = \psi_0 \ys{0} + \psi_1 \ys{1} $  is sometimes described as "0 and 1 at the
same time", although it is important to realize that the notion of time plays no role here. This equation is a well-defined mathematical object that completely describes the state of a single qubit.
\subsection*{Quantum gate}
A quantum gate is a unitary operation that can be performed on a qubit. All single-qubit quantum gates can be visualized as rotations of $ \ys{\psi} $. Quantum gates are represented by unitary matrices. A gate which acts on $n$ qubits is represented by a $2^{n}\times 2^{n} $ unitary matrix, and the set of all such gates with the group operation of matrix multiplication is the symmetry group $U(2^n)$. The quantum states that the gates act upon are unit vectors in $2^{n}$ complex dimensions, with the complex Euclidean norm. The basis vectors (sometimes called eigenstates) are the possible outcomes if measured, and a quantum state is a linear combination of these outcomes. The most common quantum gates operate on vector spaces of one or two qubits, just like the common classical logic gates operate on one or two bits. 


The Pauli gates $(X,Y,Z)$ are the three Pauli matrices $(\sigma _{x},\sigma _{y},\sigma _{z})$ and act on a single qubit. The Pauli X, Y and Z equate to a rotation around the x, y and z axes of the Bloch sphere by $\pi$  radians. These matrices are usually represented as 
\begin{align*}
    &X = \sigma_x = \begin{bmatrix}
        0 & 1 \\ 1 & 0
    \end{bmatrix} \\
    &Y = \sigma_y = \begin{bmatrix}
        0 & -i \\ i & 0
    \end{bmatrix} \\
    &Z = \sigma_z = \begin{bmatrix}
        1 & 0 \\ 0 & -1
    \end{bmatrix}
\end{align*}


There are a lot of other quantum gates such as controlled gates, phase shift gates, Hadamard gate swap gate, all of which  are the basic of the gate-based quantum computing. 


\section*{Different categories of quantum computer}
\subsection*{Analog quantum computer}
This type of system performs its operation by manipulating the analog values in the Hamiltonian representation. It does not use quantum gates. It includes quantum annealing, quantum simulation and adiabatic quantum computing. The quantum annealing is done using some initial set of qubits that gradually changes the energy encountered by the system until the problem parameters are defined by Hamiltonian. This is done in order to get the highest probability final state of the qubits that corresponds to the solution of that problem. The adiabatic quantum computer performs computation using some initial set of qubits in the Hamiltonian ground state and then Hamiltonian is changed slowly enough such that it stays in its ground state or lowest possible energy while the process takes place. It has processing power similar to a gate-based computer but still cannot perform full error correction.

\subsection*{NISQ gate-based computer}
NISQ stands for Noisy Intermediate-Scale Quantum. It is also known as the Digital NISQ computer. These type of systems are gate-based and operates on a collection of qubits without full error correction and cannot restrict all the errors. The computations must be designed in a way so that they remain practical on a quantum system with little noise and can be finished in fewer and sufficient steps so that Decoherence and gate errors do not hide the outcomes.

\subsection*{Gate-based quantum computer with full error correction}
Such computers also perform gate-based operations on a set of qubits with the implementation of the Quantum Error Correction algorithm. It reduces or corrects the noise in the system occurring during the computation period. Errors may include inadequate signals, device forgery or undesired bonding of qubits to the environment or with each other. The error is reduced to such a limit that the system seems valid and precise for all computations. Such quantum computers can have various realizations and they must fulfill some conditions such as there must be an availability of a well-defined two-level system that can be used as qubits, a potential to initialize those qubits, a sufficiently extended amount of Decoherence time which can perform error correction and computation, quantum gates (a set of quantum operations) common for every quantum computation and a capability of measuring each quantum bit individually without bothering others. The analog quantum computers and digital NISQ computers are in progress while the gate-based computers with full error corrections are much more difficult and demanding.

\section*{Applications of quantum computing}
Many quantum algorithms have been evolved for quantum computers that deliver speedup which is a result of some fundamental mathematical methods like Fourier transform, Hamiltonian simulation, etc. Most algorithms require a large number of qubits of the best quality and some error correction to provide useful functionalities. These algorithms are formed in blocks rather than as a whole combined application since it is not practical. Therefore, it is a great challenge to create quantum applications that are really practically useful along with providing speedup with no error. The potential utility or say useful application of a quantum computer is an area of ongoing research. It is predicted that those applications require fewer qubits and can be carried out with a lesser amount of codes. It is possible to build algorithms that can run faster on quantum computers because of the distinct features of the qubit. Below are some of the primary applications that we will see soon in the upcoming era:
\subsection*{Optimization Problems}
Optimizing a problem implies finding the best solution to that problem out of all the possible solutions. It can be done by minimizing the error and even minimizing the steps available. Quantum computers are best in solving optimization problems. There are a lot of quantum algorithms out of which quantum optimization algorithms might improve the already existing optimization problems which are solved using conventional computers currently. Some of them are quantum semi-definite programming, quantum data fitting, and quantum combinatorial optimization. Some of the examples include simulating the molecular model like protein behavior for medical research which can lead to the new discovery of drugs for serious diseases like cancer, lung disease, etc. Another example is the Simulation of the cellular structure of batteries for improving battery power and life in electric vehicles. It could also solve travel-related problems in real traffic just like traveling salesman problems to find the shortest path between many cities, going to each city once and returning back, modeling the entire finance market, and many more. Traveling optimization is the major work under Volkswagen recently.
\subsection*{Artificial Intelligence}
Artificial Intelligence counts on processing large and complex datasets. It is responsible for learning, inferring, and understanding. It learns until it stops mistaking and making errors in its task. It takes a significant amount of time in learning too. But quantum computing can make it easy and more accurate. Since conventional computers are only training the learning model from a specific size of the dataset to restrict the computation time. Quantum computers can train these models over a huge dataset without sticking into the exponential time. The more data it uses to train, the more accurate it will be. Generative models generate output such as image, audio, etc. that can be fed to quantum computers to improve its quality and accuracy. Natural Language processing is another example that can understand complete sentences. Quantum computers can make it understand all the phrases and speech in real-time with improved quality, which is computationally costly with today’s computer.
\subsection*{Quantum Simulation}
It is an important utility in the field of quantum chemistry and material science [31]. This problem needs solving ground state energies of electrons and their wave functions, with or without the presence of some external electric or magnetic field. From the structure of atoms and electrons in chemistry to the rate at which chemical reactions are taking place, everything can be simulated very well. The classical computer when applied to this problem often fails to reach the level of precision needed to predict the rate of the chemical reaction.


It could also have commercial applications in areas such as medical and healthcare fields, chemical catalysts, storage of energy, pharmaceutical advancement and device displays.
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